Coefficient of $x^{3}$ and $x^{4}$ in ${(1 + x + x^{2} + x^{3} + x^{4})}^{199} {(x - 1)}^{201}$ are .

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Solution

$S = (1 + x + x^{2} + x^{3} + x^{4})^{199} (x - 1)^{201}$
$= (\frac{x^{5} - 1}{x - 1})^{199} (x - 1)^{201}$
$= (x^{5} - 1)^{199} (x - 1)^{2}$
$= (1 - 2 x + x^{2}) (x^{5} - 1)^{199}$
$= (x^{5} - 1)^{199} - 2 x (x^{5} - 1)^{199} + x^{2} (x^{5} - 1)^{199}$ .
Hence there will be no term of $x^{3}$ and $x^{4}$ .

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Coefficient of x3 and x4 in (1+x+x2+x3+x4)199(x−1)201 are .

S=(1+x+x2+x3+x4)199(x−1)201=(x5−1x−1)199(x−1)201=(x5−1)199(x−1)2=(1−2x+x2)(x5−1)199=(x5−1)199−2x(x5−1)199+x2(x5−1)199.Hence there will be no term of x3 and x4.

Coefficient of $x^{3}$ and $x^{4}$ in ${(1 + x + x^{2} + x^{3} + x^{4})}^{199} {(x - 1)}^{201}$ are .